tree queries part I

src/SUMMARY.md

@@ -90,9 +90,9 @@
     - [Strong connectivity](strong_connectivity.md)
         - [Kosaraju’s algorithm](kosaraju_s_algorithm.md)
         - [2SAT problem](2sat_problem.md)
-<!--     - [Tree queries](README.md) -->
-<!--         - [Finding ancestors](README.md) -->
-<!--         - [Subtrees and paths](README.md) -->
+    - [Tree queries](tree_queries.md)
+        - [Finding ancestors](finding_ancestors.md)
+        - [Subtrees and paths](subtrees_and_paths.md)
 <!--         - [Lowest common ancestor](README.md) -->
 <!--         - [Offline algorithms](README.md) -->
 <!--     - [Paths and circuits](README.md) -->

src/finding_ancestors.md

@@ -0,0 +1,58 @@
+# Finding ancestors
+
+The $k$th **ancestor** of a node $x$ in a rooted tree
+is the node that we will reach if we move $k$
+levels up from $x$.
+Let `ancestor}(x,k)` denote the $k$th ancestor of a node $x$
+(or $0$ if there is no such an ancestor).
+For example, in the following tree,
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (1) at (0,3) {1};
+\node[draw, circle] (2) at (2,1) {2};
+\node[draw, circle] (3) at (-2,1) {4};
+\node[draw, circle] (4) at (0,1) {5};
+\node[draw, circle] (5) at (2,-1) {6};
+\node[draw, circle] (6) at (-3,-1) {3};
+\node[draw, circle] (7) at (-1,-1) {7};
+\node[draw, circle] (8) at (-1,-3) {8};
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (1) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (2) -- (5);
+\path[draw,thick,-] (3) -- (6);
+\path[draw,thick,-] (3) -- (7);
+\path[draw,thick,-] (7) -- (8);
+
+\path[draw=red,thick,->,line width=2pt] (8) edge [bend left] (3);
+\path[draw=red,thick,->,line width=2pt] (2) edge [bend right] (1);
+\end{tikzpicture}
+</script>
+
+An easy way to calculate any value of `ancestor(x,k)`
+is to perform a sequence of $k$ moves in the tree.
+However, the time complexity of this method
+is $O(k)$, which may be slow, because a tree of $n$
+nodes may have a chain of $n$ nodes.
+
+Fortunately, using a technique similar to that
+used in Chapter 16.3, any value of `ancestor(x,k)`
+can be efficiently calculated in $O(\log k)$ time
+after preprocessing.
+The idea is to precalculate all values `ancestor(x,k)`
+where $k \le n$ is a power of two.
+For example, the values for the above tree
+are as follows:
+
+| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
+| - | - | - | - | - | - | - | - | - |
+| `ancestor(x,1)` | 0 | 1 | 4 | 1 | 1 | 2 | 4 | 7 |
+| `ancestor(x,2)` | 0 | 0 | 1 | 0 | 0 | 1 | 1 | 4 |
+| `ancestor(x,4)` | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
+
+The preprocessing takes $O(n \log n)$ time,
+because $O(\log n)$ values are calculated for each node.
+After this, any value of `ancestor(x,k)` can be calculated
+in $O(\log k)$ time by representing $k$
+as a sum where each term is a power of two.

src/subtrees_and_paths.md

@@ -0,0 +1,307 @@
+# Subtrees and paths
+
+A **tree traversal array** contains the nodes of a rooted tree
+in the order in which a depth-first search
+from the root node visits them.
+For example, in the tree
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (1) at (0,3) {1};
+\node[draw, circle] (2) at (-3,1) {2};
+\node[draw, circle] (3) at (-1,1) {3};
+\node[draw, circle] (4) at (1,1) {4};
+\node[draw, circle] (5) at (3,1) {5};
+\node[draw, circle] (6) at (-3,-1) {6};
+\node[draw, circle] (7) at (-0.5,-1) {7};
+\node[draw, circle] (8) at (1,-1) {8};
+\node[draw, circle] (9) at (2.5,-1) {9};
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (1) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (1) -- (5);
+\path[draw,thick,-] (2) -- (6);
+\path[draw,thick,-] (4) -- (7);
+\path[draw,thick,-] (4) -- (8);
+\path[draw,thick,-] (4) -- (9);
+\end{tikzpicture}
+</script>
+
+a depth-first search proceeds as follows:
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (1) at (0,3) {1};
+\node[draw, circle] (2) at (-3,1) {2};
+\node[draw, circle] (3) at (-1,1) {3};
+\node[draw, circle] (4) at (1,1) {4};
+\node[draw, circle] (5) at (3,1) {5};
+\node[draw, circle] (6) at (-3,-1) {6};
+\node[draw, circle] (7) at (-0.5,-1) {7};
+\node[draw, circle] (8) at (1,-1) {8};
+\node[draw, circle] (9) at (2.5,-1) {9};
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (1) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (1) -- (5);
+\path[draw,thick,-] (2) -- (6);
+\path[draw,thick,-] (4) -- (7);
+\path[draw,thick,-] (4) -- (8);
+\path[draw,thick,-] (4) -- (9);
+
+
+\path[draw=red,thick,->,line width=2pt] (1) edge [bend right=15] (2);
+\path[draw=red,thick,->,line width=2pt] (2) edge [bend right=15] (6);
+\path[draw=red,thick,->,line width=2pt] (6) edge [bend right=15] (2);
+\path[draw=red,thick,->,line width=2pt] (2) edge [bend right=15] (1);
+\path[draw=red,thick,->,line width=2pt] (1) edge [bend right=15] (3);
+\path[draw=red,thick,->,line width=2pt] (3) edge [bend right=15] (1);
+\path[draw=red,thick,->,line width=2pt] (1) edge [bend right=15] (4);
+\path[draw=red,thick,->,line width=2pt] (4) edge [bend right=15] (7);
+\path[draw=red,thick,->,line width=2pt] (7) edge [bend right=15] (4);
+\path[draw=red,thick,->,line width=2pt] (4) edge [bend right=15] (8);
+\path[draw=red,thick,->,line width=2pt] (8) edge [bend right=15] (4);
+\path[draw=red,thick,->,line width=2pt] (4) edge [bend right=15] (9);
+\path[draw=red,thick,->,line width=2pt] (9) edge [bend right=15] (4);
+\path[draw=red,thick,->,line width=2pt] (4) edge [bend right=15] (1);
+\path[draw=red,thick,->,line width=2pt] (1) edge [bend right=15] (5);
+\path[draw=red,thick,->,line width=2pt] (5) edge [bend right=15] (1);
+
+\end{tikzpicture}
+</script>
+
+Hence, the corresponding tree traversal array is as follows:
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.7]
+\draw (0,0) grid (9,1);
+
+\node at (0.5,0.5) {1};
+\node at (1.5,0.5) {2};
+\node at (2.5,0.5) {6};
+\node at (3.5,0.5) {3};
+\node at (4.5,0.5) {4};
+\node at (5.5,0.5) {7};
+\node at (6.5,0.5) {8};
+\node at (7.5,0.5) {9};
+\node at (8.5,0.5) {5};
+% 
+% \footnotesize
+% \node at (0.5,1.4) {1};
+% \node at (1.5,1.4) {2};
+% \node at (2.5,1.4) {3};
+% \node at (3.5,1.4) {4};
+% \node at (4.5,1.4) {5};
+% \node at (5.5,1.4) {6};
+% \node at (6.5,1.4) {7};
+% \node at (7.5,1.4) {8};
+% \node at (8.5,1.4) {9};
+\end{tikzpicture}
+</script>
+
+## Subtree queries
+
+Each subtree of a tree corresponds to a subarray
+of the tree traversal array such that
+the first element of the subarray is the root node.
+For example, the following subarray contains the
+nodes of the subtree of node $4$:
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.7]
+\fill[color=lightgray] (4,0) rectangle (8,1);
+\draw (0,0) grid (9,1);
+
+\node at (0.5,0.5) {1};
+\node at (1.5,0.5) {2};
+\node at (2.5,0.5) {6};
+\node at (3.5,0.5) {3};
+\node at (4.5,0.5) {4};
+\node at (5.5,0.5) {7};
+\node at (6.5,0.5) {8};
+\node at (7.5,0.5) {9};
+\node at (8.5,0.5) {5};
+% 
+% \footnotesize
+% \node at (0.5,1.4) {1};
+% \node at (1.5,1.4) {2};
+% \node at (2.5,1.4) {3};
+% \node at (3.5,1.4) {4};
+% \node at (4.5,1.4) {5};
+% \node at (5.5,1.4) {6};
+% \node at (6.5,1.4) {7};
+% \node at (7.5,1.4) {8};
+% \node at (8.5,1.4) {9};
+\end{tikzpicture}
+</script>
+
+Using this fact, we can efficiently process queries
+that are related to subtrees of a tree.
+As an example, consider a problem where each node
+is assigned a value, and our task is to support
+the following queries:
+
+- update the value of a node
+- calculate the sum of values in the subtree of a node
+
+Consider the following tree where the blue numbers
+are the values of the nodes.
+For example, the sum of the subtree of node $4$
+is $3+4+3+1=11$.
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.9]
+\node[draw, circle] (1) at (0,3) {1};
+\node[draw, circle] (2) at (-3,1) {2};
+\node[draw, circle] (3) at (-1,1) {3};
+\node[draw, circle] (4) at (1,1) {4};
+\node[draw, circle] (5) at (3,1) {5};
+\node[draw, circle] (6) at (-3,-1) {6};
+\node[draw, circle] (7) at (-0.5,-1) {7};
+\node[draw, circle] (8) at (1,-1) {8};
+\node[draw, circle] (9) at (2.5,-1) {9};
+
+\path[draw,thick,-] (1) -- (2);
+\path[draw,thick,-] (1) -- (3);
+\path[draw,thick,-] (1) -- (4);
+\path[draw,thick,-] (1) -- (5);
+\path[draw,thick,-] (2) -- (6);
+\path[draw,thick,-] (4) -- (7);
+\path[draw,thick,-] (4) -- (8);
+\path[draw,thick,-] (4) -- (9);
+
+\node[color=blue] at (0,3+0.65) {2};
+\node[color=blue] at (-3-0.65,1) {3};
+\node[color=blue] at (-1-0.65,1) {5};
+\node[color=blue] at (1+0.65,1) {3};
+\node[color=blue] at (3+0.65,1) {1};
+\node[color=blue] at (-3,-1-0.65) {4};
+\node[color=blue] at (-0.5,-1-0.65) {4};
+\node[color=blue] at (1,-1-0.65) {3};
+\node[color=blue] at (2.5,-1-0.65) {1};
+\end{tikzpicture}
+</script>
+
+The idea is to construct a tree traversal array that contains
+three values for each node: the identifier of the node,
+the size of the subtree, and the value of the node.
+For example, the array for the above tree is as follows:
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.7]
+\draw (0,1) grid (9,-2);
+
+\node[left] at (-1,0.5) {node id};
+\node[left] at (-1,-0.5) {subtree size};
+\node[left] at (-1,-1.5) {node value};
+
+\node at (0.5,0.5) {1};
+\node at (1.5,0.5) {2};
+\node at (2.5,0.5) {6};
+\node at (3.5,0.5) {3};
+\node at (4.5,0.5) {4};
+\node at (5.5,0.5) {7};
+\node at (6.5,0.5) {8};
+\node at (7.5,0.5) {9};
+\node at (8.5,0.5) {5};
+
+\node at (0.5,-0.5) {9};
+\node at (1.5,-0.5) {2};
+\node at (2.5,-0.5) {1};
+\node at (3.5,-0.5) {1};
+\node at (4.5,-0.5) {4};
+\node at (5.5,-0.5) {1};
+\node at (6.5,-0.5) {1};
+\node at (7.5,-0.5) {1};
+\node at (8.5,-0.5) {1};
+
+\node at (0.5,-1.5) {2};
+\node at (1.5,-1.5) {3};
+\node at (2.5,-1.5) {4};
+\node at (3.5,-1.5) {5};
+\node at (4.5,-1.5) {3};
+\node at (5.5,-1.5) {4};
+\node at (6.5,-1.5) {3};
+\node at (7.5,-1.5) {1};
+\node at (8.5,-1.5) {1};
+% 
+% \footnotesize
+% \node at (0.5,1.4) {1};
+% \node at (1.5,1.4) {2};
+% \node at (2.5,1.4) {3};
+% \node at (3.5,1.4) {4};
+% \node at (4.5,1.4) {5};
+% \node at (5.5,1.4) {6};
+% \node at (6.5,1.4) {7};
+% \node at (7.5,1.4) {8};
+% \node at (8.5,1.4) {9};
+\end{tikzpicture}
+</script>
+
+Using this array, we can calculate the sum of values
+in any subtree by first finding out the size of the subtree
+and then the values of the corresponding nodes.
+For example, the values in the subtree of node $4$
+can be found as follows:
+
+<script type="text/tikz">
+\begin{tikzpicture}[scale=0.7]
+\fill[color=lightgray] (4,1) rectangle (5,0);
+\fill[color=lightgray] (4,0) rectangle (5,-1);
+\fill[color=lightgray] (4,-1) rectangle (8,-2);
+\draw (0,1) grid (9,-2);
+
+\node[left] at (-1,0.5) {node id};
+\node[left] at (-1,-0.5) {subtree size};
+\node[left] at (-1,-1.5) {node value};
+
+\node at (0.5,0.5) {1};
+\node at (1.5,0.5) {2};
+\node at (2.5,0.5) {6};
+\node at (3.5,0.5) {3};
+\node at (4.5,0.5) {4};
+\node at (5.5,0.5) {7};
+\node at (6.5,0.5) {8};
+\node at (7.5,0.5) {9};
+\node at (8.5,0.5) {5};
+
+\node at (0.5,-0.5) {9};
+\node at (1.5,-0.5) {2};
+\node at (2.5,-0.5) {1};
+\node at (3.5,-0.5) {1};
+\node at (4.5,-0.5) {4};
+\node at (5.5,-0.5) {1};
+\node at (6.5,-0.5) {1};
+\node at (7.5,-0.5) {1};
+\node at (8.5,-0.5) {1};
+
+\node at (0.5,-1.5) {2};
+\node at (1.5,-1.5) {3};
+\node at (2.5,-1.5) {4};
+\node at (3.5,-1.5) {5};
+\node at (4.5,-1.5) {3};
+\node at (5.5,-1.5) {4};
+\node at (6.5,-1.5) {3};
+\node at (7.5,-1.5) {1};
+\node at (8.5,-1.5) {1};
+% 
+% \footnotesize
+% \node at (0.5,1.4) {1};
+% \node at (1.5,1.4) {2};
+% \node at (2.5,1.4) {3};
+% \node at (3.5,1.4) {4};
+% \node at (4.5,1.4) {5};
+% \node at (5.5,1.4) {6};
+% \node at (6.5,1.4) {7};
+% \node at (7.5,1.4) {8};
+% \node at (8.5,1.4) {9};
+\end{tikzpicture}
+</script>
+
+To answer the queries efficiently,
+it suffices to store the values of the
+nodes in a binary indexed or segment tree.
+After this, we can both update a value
+and calculate the sum of values in $O(\log n)$ time.

src/tree_queries.md

@@ -0,0 +1,11 @@
+# Tree queries
+
+This chapter discusses techniques for
+processing queries on
+subtrees and paths of a rooted tree.
+For example, such queries are:
+
+- what is the $k$th ancestor of a node?
+- what is the sum of values in the subtree of a node?
+- what is the sum of values on a path between two nodes?
+- what is the lowest common ancestor of two nodes?